Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{x^2 - 8x + 12}{x^2 - 11x + 30}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 - 8x + 12}{x^2 - 11x + 30} = \dfrac{(x - 2)(x - 6)}{(x - 5)(x - 6)} $ Notice that the term $(x - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 6)$ gives: $a = \dfrac{x - 2}{x - 5}$ Since we divided by $(x - 6)$, $x \neq 6$. $a = \dfrac{x - 2}{x - 5}; \space x \neq 6$